Learn Turkish: Onun buluşu herkese ilham oldu. - Her discovery was an inspiration to everyone.

It comes from the verb bulmak (to find). By adding the suffix -uş (which forms a noun from a verb), it becomes buluş, meaning "a discovery" or "an invention". The final -u in buluşu is the possessive suffix linking it to Onun (her).

Why is there an -e at the end of herkese?

The phrase ilham olmak (to be an inspiration) takes the dative case for the person receiving the inspiration. Just as in English you might say "was an inspiration to everyone," Turkish uses the dative suffix -e on herkes (everyone) to make herkese.

Could I say ilham verdi instead of ilham oldu here?

Absolutely. İlham vermek (to give inspiration) is just as common as ilham olmak (to be/become an inspiration). Both require the dative case, so Onun buluşu herkese ilham verdi is perfectly natural and means the same thing.

Since this lesson is about the escalating word hatta, how could I add it to this sentence?

You could place hatta at the very beginning to escalate a previous statement. For example, if someone previously said "She worked very hard," you could add: "Hatta, onun buluşu herkese ilham oldu." (In fact / What's more, her discovery was an inspiration to everyone.)